Optimal. Leaf size=98 \[ \frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)} \]
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Rubi [A]
time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2215, 2221,
2317, 2438} \begin {gather*} -\frac {d \text {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{a f g n \log (F)}+\frac {(c+d x)^2}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2317
Rule 2438
Rubi steps
\begin {align*} \int \frac {c+d x}{a+b \left (F^{g (e+f x)}\right )^n} \, dx &=\frac {(c+d x)^2}{2 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{a+b \left (F^{g (e+f x)}\right )^n} \, dx}{a}\\ &=\frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}+\frac {d \int \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right ) \, dx}{a f g n \log (F)}\\ &=\frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}+\frac {d \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f^2 g^2 n^2 \log ^2(F)}\\ &=\frac {(c+d x)^2}{2 a d}-\frac {(c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f g n \log (F)}-\frac {d \text {Li}_2\left (-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a f^2 g^2 n^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 74, normalized size = 0.76 \begin {gather*} \frac {-f g n (c+d x) \log (F) \log \left (1+\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )+d \text {Li}_2\left (-\frac {a \left (F^{g (e+f x)}\right )^{-n}}{b}\right )}{a f^2 g^2 n^2 \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(678\) vs.
\(2(96)=192\).
time = 0.04, size = 679, normalized size = 6.93
method | result | size |
risch | \(\frac {c \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g n a}-\frac {c \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g n a}+\frac {d \,x^{2}}{a}+\frac {d x e}{f a}+\frac {d x \left (\ln \left (F^{g \left (f x +e \right )}\right )-g \left (f x +e \right ) \ln \left (F \right )\right )}{\ln \left (F \right ) f g a}-\frac {d \ln \left (F^{g \left (f x +e \right )}\right ) x}{\ln \left (F \right ) f g a}+\frac {d \ln \left (F^{g \left (f x +e \right )}\right )^{2}}{2 \ln \left (F \right )^{2} f^{2} g^{2} a}-\frac {d \ln \left (1+\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right ) x}{\ln \left (F \right ) f g n a}-\frac {d \ln \left (1+\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right ) e}{\ln \left (F \right ) f^{2} g n a}-\frac {d \ln \left (1+\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right ) \left (\ln \left (F^{g \left (f x +e \right )}\right )-g \left (f x +e \right ) \ln \left (F \right )\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n a}-\frac {d \polylog \left (2, -\frac {b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}}{a}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n^{2} a}-\frac {d e \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f^{2} g n a}+\frac {d e \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f^{2} g n a}-\frac {d \left (\ln \left (F^{g \left (f x +e \right )}\right )-g \left (f x +e \right ) \ln \left (F \right )\right ) \ln \left (F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n a}+\frac {d \left (\ln \left (F^{g \left (f x +e \right )}\right )-g \left (f x +e \right ) \ln \left (F \right )\right ) \ln \left (a +b \,F^{n g f x} F^{-n g f x} \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right )^{2} f^{2} g^{2} n a}\) | \(679\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 155, normalized size = 1.58 \begin {gather*} -\frac {2 \, {\left (c f g n - d g n e\right )} \log \left (F^{f g n x + g n e} b + a\right ) \log \left (F\right ) - {\left (d f^{2} g^{2} n^{2} x^{2} + 2 \, c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} + 2 \, {\left (d f g n x + d g n e\right )} \log \left (F\right ) \log \left (\frac {F^{f g n x + g n e} b + a}{a}\right ) + 2 \, d {\rm Li}_2\left (-\frac {F^{f g n x + g n e} b + a}{a} + 1\right )}{2 \, a f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c + d x}{a + b \left (F^{e g} F^{f g x}\right )^{n}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {c+d\,x}{a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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